# from local volatility function to implied volatility for a given strike

Assuming we know the local volatility function σ(S,t) for all S and t, how can we recover the corresponding implied volatility for a given strike K (and the other necessary parameters, S, r, T, t...)? What is the formal mathematical way to express the dependency σimplied(K) to the σ(S,t)s?

Can you also recommend documents where I can find numerical methodology?

• Look for approximations based on the "most likely path" (Gatheral, Reghai etc.) – Quantuple Jan 18 '17 at 7:20
• Alternatively, you could explicitly value the corresponding European plain vanilla options. E.g. use finite differences to solve the local volatility pricing PDE and then compute their implied volatilities. – LocalVolatility Jan 18 '17 at 15:25